Virtual knot theory on a group

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Virtual knot theory on a group

Arnaud Mortier

To cite this version:

Arnaud Mortier. Virtual knot theory on a group. 2014. �hal-00958872�

Virtual knot theory on a group

Arnaud Mortier arno.mortier@laposte.net

March 13, 2014

Abstract

Given a group endowed with a Z/2-valued morphism we associate a Gauss diagram theory, and show that for a particular choice of the group these diagrams encode faithfully virtual knots on a given arbitrary surface. This theory contains all of the earlier attempts to decorate Gauss diagrams, in a way that is made precise viasymmetry-preserving maps. These maps become crucial when one makes use of decorated Gauss diagrams to describe finite-type invariants. In particular they allow us to generalize Grishanov-Vassiliev’s formulas and to show that they define invariants of virtual knots.

Contents

1 Preliminary: classical Gauss diagrams and their Reidemeister moves 2 2 Knot and virtual knot diagrams on an arbitrary surface 4

2.1 Thickenings of surfaces . . . 4

2.2 Diagram isotopies and detour moves . . . 5

3 Virtual knot theory on a weighted group 6 3.1 General settings and the main theorem . . . 7

3.1.1 About the orbits ofw-moves . . . 11

3.2 Abelian Gauss diagrams . . . 11

3.3 Homological formulas . . . 15

3.3.1 The energy formula . . . 15

3.3.2 The torsion formula . . . 16

4 Finite-type invariants 19 4.1 General algebraic settings . . . 20

4.1.1 The Polyak algebra . . . 20

4.1.2 The symmetry-preserving injections . . . 21

4.1.3 Arrow diagrams and homogeneous invariants . . . 23

4.1.4 Based and degenerate diagrams . . . 25

4.2 Invariance criteria . . . 27

4.2.1 Invariance criterion forw-orbits . . . . 28

4.3 Examples and applications . . . 30

4.3.1 Grishanov-Vassiliev’s planar chain invariants . . . 30

4.3.2 There is a Whitney index for non nullhomotopic virtual knots . . . 32 Gauss diagrams were introduced in knot theory as a means of representing knots and their finite- type invariants [20, 9], allowing compactification and generalization of formulas due to J.Lannes [14]. Since then, several generalizations have been attempted to adapt them to knot theory in thickened surfaces by decorating them with topological information [7, 11, 17].

Our goal is to construct a unifying “father” framework, and to describe how to get down from there to other versions with less data.

First we define and study (virtual) knot diagrams on an arbitrary surface Σ: these are tetrava- lent graphs embedded in Σ, some of whose double points (the “real” ones) are pushed and desin- gularized into a real line bundle over Σ. Defining Gauss diagrams requires a global notion for the

branches at a real crossing to be one “over” the other, and a global notion of writhe of a crossing.

It is shown that these notions can be defined simultaneously if and only if Σ is orientable. If it is not, we sacrifice the globality of one property, and take into account its monodromy. It is shown that when the total space of the bundle is orientable, the writhes are globally defined and the monodromy of the “over/under” datum is the first Stiefel-Whitney class of the tangent bundle to Σ,w1(Σ).

In Section 3 is given a definition of Gauss diagrams decorated by elements of a fixed groupπ, subject to usual Reidemeister moves, and to additional “conjugacy moves”, depending on a fixed group homomorphismw:π→F₂. It is shown that when there is a surface Σ such thatπ=π(Σ) andw1=w1(Σ), then there is a 1−1 correspondence between Gauss diagrams and virtual knot diagrams, that induces a correspondence between the equivalence classes (virtual knot types) on both sides.

A lighter kind of Gauss diagrams, calledabelian, is defined in Subsection 3.2 following the idea of T.Fiedler’sH1(Σ)-decorated diagrams ([7]) and shown to be equivalent to the above whenπis abelian andw is trivial. The little drawback of this version is that it becomes more difficult to compute the homological decoration of an arbitrary loop. Two formulas are presented in 3.3 to sort this out, involving quite unexpected combinatorial tools.

Finally, we describe invariance criteria for the analog of Goussarov-Polyak-Viro’s invariants [10]

in this framework. As an application, we obtain a generalization of Grishanov-Vassiliev formulas [11], and a notion of Whitney index for virtual knots whose underlying immersed curve is non nullhomotopic.

Acknowledgements

I wish to thank Christian Blanchet who invited me at the Institut de Mathématiques de Jussieu where this work was done. This work has benefited from discussions with Michael Polyak, Micah Chrisman, and Thomas Fiedler.

1 Preliminary: classical Gauss diagrams and their Reide- meister moves

Definition 1.1. Aclassical Gauss diagram is an equivalence class of an oriented circle in which a finite number of couples of points are linked by an abstract oriented arrow with a sign decoration, up to positive homeomorphism of the circle. A Gauss diagram withnarrows is said to be ofdegree n.

It may happen that one regards Gauss diagrams as topological objects (drawing loops on them, considering their first homology). In that case, one must beware of the fact thatthe arrows do not topologically intersect – that is what is meant by “abstract”. However, the fact that two arrows maylook like they intersect is something combinatorially well-defined, and interesting for many purposes.

Fact: There is a natural way to associate a Gauss diagram with a knot diagram in the sphere S², from which the knot diagram can be uniquely recovered. Fig.1 illustrates this fact.

a c d

a a b b

c

c d

d

b −

+

− + +

−

Figure 1: The writhe convention, a diagram of the figure eight knot, and its Gauss diagram – the letters are here only for the sake of clarity.

However, not every Gauss diagram actually comes from a knot diagram in that way. This observation has lead to the development of virtual knot theory [13]: basically a virtual knot is a Gauss diagram which does not come from an actual knot. There is a knot-diagrammatic version of these, using virtual crossings subject to virtual Reidemeister moves - that can be thought of as a unique “detour move”. A detour move is naturally any move that leaves the underlying Gauss diagram unchanged.

Of course virtual knot diagrams are also subject to the usual Reidemeister moves, and these do change the face of the Gauss diagram. We call them R-moves for simplicity - and to make it clear whether knot diagrams or Gauss diagrams are considered. Here is a combinatorial description of R-moves.

R₁-moves

An R1-move is the birth or death of an isolated arrow, as shown in Fig.2 (top-left). There is no restriction on the direction or the sign of the arrow.

R2-moves

An R2-move is the birth or death of a pair of arrows with different signs, whose heads are consecutive as well as their tails (Fig.2, top-right).

If one restricts oneself to Gauss diagrams that come from classical knot diagrams, then there is an additional condition as for the creating direction: indeed, two arcs in a knot diagram can be subject to a Reidemeister II move if and only if theyface each other. In the virtual world, there is no such condition since any two arcs can be brought to face each other by detour moves.

It may be good to know that this condition can be read directly on the Gauss diagram: indeed, two arcs face each other in a knot diagram if one can join them by walking along the diagram and turning to the left at each time one meets a crossing. Thanks to the decorations of the arrows, it makes sense for a path in a Gauss diagram toturn to the left.

R 1 R

2

R 3

=

= =

=

=

Figure 2: R-moves for Gauss diagrams (see above and below the rules for the decorations)

R₃-moves

Definition 1.2. In a classical Gauss diagram of degree n, the complementary of the arrows is made of 2noriented components. These are called theedges of the diagram. In a diagram with no arrow, we still call the whole circle an edge.

Letebe an edge in a Gauss diagram, between two consecutive arrow ends that do not belong to the same arrow. Put

η(e) =

+1 if the arrows that boundecross each other

−1 otherwise ,

and let↑(e) be the number of arrowheads at the boundary ofe. Then define ε(e) =η(e)·(−1)^↑(e).

Finally, define w(e) as the product of the writhes of the two arrows at the boundary ofe.

An R3-move is the simultaneous switch of the endpoints of three arrows as shown on Fig.2 (bottom), with the following conditions:

1. The value of w(e)ε(e) should be the same for all three visible edgese. This ensures that the piece of diagram containing the three arrows can be represented in a knot-diagrammatic way without making use of virtual crossings.

2. The values of↑(e) should be pairwise different. This ensures that one of the arcs in the knot diagram version actually “goes over” the others.

Remark 1.3. From a simplicial viewpoint, the sign w(e)ε(e) gives a natural co-orientation of the 1-codimensional strata corresponding to R3moves. This is exploited in [15] to construct finite-type 1-cocycles.

2 Knot and virtual knot diagrams on an arbitrary surface

The goal of this section is to examine when and how one can define a couple of equivalent theories

“virtual knots−Gauss diagrams” that generalizes knot theory in an arbitrary 3-manifoldM. What first appears is that a Gauss diagram depends on a projection; so it seems unavoidable to ask for the existence of a surface Σ (maybe with boundary, non orientable, or non compact), and a “nice”

mapp:M →Σ. For theover andunder branches at a crossing to be well-defined at least locally, the fibers ofpneed to be equipped with a total order: this leaves only the possiblity of a real line bundle.

2.1 Thickenings of surfaces

Let us now split the discussion according to the two kinds of decorations that one would expect to find on a Gauss diagram: signs (local writhes), and orientation of the arrows.

Local writhes

For a knot in an arbitrary real line bundle, there are situations in which it is possible to switch over and under in a crossing by a merediagram isotopy. For instance, in the non-trivial line bundle over the annulusS¹×R, a full rotation of the closure of the two-stranded elementary braidσ1 turns it into the closure ofσ₁⁻¹ (Fig. 3).

P P

P P

Figure 3: Non trivial line bundle over the annulus – as one reads from left to right, the knot moves towards the right of the picture.

Fig. 3 would be exactly the same (except for the gluing indications) if one considered the trivial line bundle over the Moebius strip. Note that this diagram would then represent a 2-component

link. In fact, it is possible to embed this picture inany non-orientable total space of a line bundle over a surface.

This phenomenon reveals the fact that in these cases, there is no way to define the local writhe of a crossing. However, according to [6] (Definition 1.), there is a well-defined writhe as soon as the total space of the bundle is oientable.

Definition 2.1. We call athickened surfacea real line bundle over a surface, whose total space is orientable.

Definition-Lemma 2.2. If M → Σ is a thickened surface, then its first Stiefel-Whitney class coincides with that of the tangent bundle to Σ. This class induces a homomorphism w1(Σ) : π1(Σ)→F₂. The couple(π1(Σ), w1(Σ)) is called the weighted fundamental group ofΣ. Note that in particular the thickening ofΣis the trivial bundle Σ×Rif and only if Σis orientable.

Arrow orientations

Note that the writhe of a crossing for a knot inM →Σ depends only on one choice, that of an orientation for M. The important thing is that this choice is global, so that it makes sense to compare the writhes of different crossings (they live in “the same”Z/2Z).

Similarly, for the orientation of the arrows in a Gauss diagram to simultaneously make sense, one needs a global definition of the over/under datum at the crossings; that is, the fibres ofM →Σ should be simultaneously and consistently oriented. In other words,M →Σ should be the trivial line bundle.

According to our definition of a thickened surface, this happens only if the surface is orientable.

So it seems that one has a choice to make, either restricting one’s attention to orientable surfaces, or taking into account the monodromy of whatever is not globally defined. Additional conjugacy moves will be needed when one defines Gauss diagrams. The convention to consider only fibre bundles with an orientable total space is arbitrary, its only use is to reduce the number of monodromy morphisms to1instead of 2.

Virtual knot diagrams on an arbitrary surface

Fix an arbitrary surface Σ and denote its thickening byM →Σ.

Definition 2.3. Avirtual knot diagram on Σ is a generic immersionS¹→Σ whose every double point has been decorated

➺ either with the designation “virtual” (which is nothing but a name),

➺ or with a way to desingularize it locally intoM, up to local isotopy.

These diagrams are subject to the usual Reidemeister moves, dictated by local isotopy inM, and to the virtual “detour” moves which are studied in the next section. As explained before, if one chooses an orientation for M, then the real crossings of such a diagram have a well-defined writhe.

2.2 Diagram isotopies and detour moves

Here by knot diagram we mean a virtual knot diagram on a fixed arbitrary surface Σ, as defined above. In this case a diagram isotopy, usually briefly denoted by H : Id→h, is the datum of a diffeomorphismhof Σ together with an isotopy from IdΣtoh. Adetour moveis a boundary-fixing homotopy of an arc that, before and after the homotopy, goes through only virtual crossings (such an arc is called totally virtual). Though both of these processes seem rather simple, it will be useful to understand how they interact.

Lemma 2.4. A knot diagram obtained from another by a sequence of diagram isotopies alternating with detour moves may always be obtained by a single diagram isotopy followed by detour moves.

Proof. It is enough to show that a detour move dfollowed by a diagram isotopy Id→hmay be replaced with a diagram isotopy followed by a detour move (without changing the initial and final diagrams). The initial diagram is denoted byD.

Callαthe totally virtual arc that is moved by the detour move. By definition,d(α) is boundary- fixing homotopic toα, and is totally virtual too. Thus, h(d(α)) andh(α) are totally virtual and boundary-fixing homotopic to each other. Sinceh(d(D)) andh(D) differ only by these two arcs, it follows that there is a detour move takingh(D) toh(d(D)).

Now an interesting question about diagram isotopies is when two of them lead to diagrams that are equivalentunder detour moves. Here is a quite useful sufficient condition.

Definition 2.5. Let X and Y be two finite subsets of Σ with the same (positive) cardinality n.

Ageneralized braid in Σ×[0,1] based on the setsX andY is an embeddingβ of a disjoint union of segments, such that Imβ∩(Σ× {t}) has cardinalitynfor eacht, coincides withX att= 0 and withY att= 1.

Let D be a knot diagram and H a diagram isotopy. Let p1 ∈ P1, . . . , pn ∈ Pn denote little neighborhoods of the real crossings ofD, and setP =∪Pi. Then,`H(pi,·) defines a generalized braid^Hβ in Σ×[0,1] with nstrands based on the sets {p1, . . . , pn} and{h(p1), . . . , h(pn)}. The strand of a braidβ that intersects Σ× {0}atpi is denoted byβi.

Proposition 2.6. Let DandH be as above. Then, up to detour moves,h(D)only depends onD and the boundary fixing homotopy class of^Hβ.

Proof. Letγbe a maximal smooth arc ofDoutsideP (thus totally virtual). It begins at somePi

and ends at somePj (of course it may happen thatj=i). Using little arcs inside ofPi andPj to join the endpoints ofγ withpi andpj, one obtains an oriented path^Hβ_i⁻¹γ^Hβj.

The obvious retraction of Σ×[0,1] onto Σ× {1} induces a map π1(Σ×[0,1], h(P)× {1})−→π1(Σ, h(P)) that sends the classh

Hβ_i⁻¹γ^Hβj

ito [h(γ)]. Since the former class is unchanged under boundary- fixing homotopy ofγ and^Hβ, so is the latter, which proves the result.

This proposition states that the only relevant datum in a diagram isotopy of a virtual knot is the path followed by the real crossings along the isotopy, up to homotopy: the entanglement of these paths with each other or themselves does not matter. It follows that the crossings may be moved one at a time:

Corollary 2.7. Let D be a knot diagram with its real crossings numbered from 1 to n, and let H: Id→hbe a diagram isotopy. Then there is a sequence of diagram isotopies H1, . . . , Hn, such that hn. . . h1(D) coincides with h(D) up to detour moves, and such that Hi is the identity on a neighborhood of each real crossing but thei-th one.

Remark 2.8. It is to be understood that thei-th crossing ofhk. . . h1(D) ishk. . . h1(pi).

Proof. Any generalized braid is (boundary-fixing) homotopic to a braidβ⊂Σ×[0,1] such that the i-th strand is vertical before the time ⁱ⁻¹_n and vertical again after the time _nⁱ. Take such a braid β that is homotopic to ^Hβ. Any diagram isotopyH^′ such thatβ =^H′β factorizes into a product Hn. . . H1 satisfying the last required condition. The fact thathn. . . h1(D) andh(D) coincide up to detour moves is a consequence of Proposition 2.6.

3 Virtual knot theory on a weighted group

In this section, we define a new Gauss diagram theory, that depends on an arbitrary groupπand a homomorphismw:π→F₂≃Z/2Z. These two data together are called aweighted group. When (π, w) is the weighted fundamental group of a surface (Definition 2.2), this theory encodes, fully

and faithfully, virtual knot diagrams on that surface (Definition 2.3).

3.1 General settings and the main theorem

Definition 3.1. Let π be an arbitrary group and w a homomorphism from π to F₂. A Gauss diagram onπ is a classical Gauss diagram decorated with

➺ an element ofπon each edge if the diagram has at least one arrow.

➺ a single element ofπup to conjugacy if the diagram is empty.

Such diagrams are subject to the usual types of R-moves, plus an additionalconjugacy move, orw-move– the dependence on warises only there. An equivalence class modulo all these moves is called avirtual knot type on (π, w).

Asubdiagram of a Gauss diagram onπis the result of removing some of its arrows. Removing an arrow involves a merging of its (2, 3, or 4) adjacent edges, and each edge resulting from this merging should be marked with the product inπ of the former markings. If all the arrows have been removed, this product is not well-defined, but its conjugacy class is.

The notion of subdiagrams is useful to construct finite-type invariants (see Section 4), but it already allows explicit understanding of

1. The distinction between empty and non empty diagrams in the definition above.

2. The “merge multiply” principle, which is omnipresent, in particular in R-moves.

An R1-moveis the local addition or removal of an isolated arrow, surrounding an edge marked with the unit 1∈π. The markings of the affected edges must satisfy the rule indicated on Fig.4 (top-left). There are no conditions on the decorations of the arrows.

Exceptional case: If the isolated arrow is the only one in the diagram on the left, then the markingsa and b on the picture actually correspond to the same edge, and the diagram on the right, with no arrow, must be decorated by [a], the conjugacy class ofa.

R 2 _−ε R 1

ab

R 3

cd ab

ε

− ε

1 1 a b

c d ε

a 1

b

1 a

1 b

c d

1 1

1 1

1 1

1

1 1 1

1

1

Figure 4: The R-moves for Gauss diagrams on a group – the exceptional cases and the rules for the missing decorations are made precise in Definition 3.1.

An R2-move is the addition or removal of two arrows with opposite writhes and matching orientations as shown on Fig.4 (top-right). The surrounded edges must be decorated with 1, and the “merge multiply” rule should be satisfied.

Exceptional case of type 1: If the markingsaand d(resp. b andc) correspond to the same edge, then the resulting marking shall becab(resp. abd).

Exceptional case of type 2: If the middle diagram contains no arrow at all,i.e. aanddmatch and so dobandc, then the (only) marking of the middle diagram shall be [ab].

−1

b a g bg⁻¹ g ag⁻¹

b a

cg ⁻¹

c d cg

−1

c a ag b

g ag−1

g b g b

g d

Figure 5: The general conjugacy move (top-left) and its two exceptional cases – in every case the orientation of the arrow switches if and only ifw(g) =−1.

An R3-movemay be of the two types shown on Fig.4 (bottom left and right). The surrounded edges must be decorated by 1, the value of w(·)ε(·) must be the same for all three of them, and the values of↑(·) must be pairwise distinct (see Definition 1.2).

A conjugacy move depends on an element g ∈π. It changes the markings of the adjacent edges to an arbitrary arrow as indicated on Fig.5. Besides, if w(g) =−1 then the orientation of the arrow is reversed – though its sign remains the same.

Remark 3.2. By composing R-moves and w-moves, it is possible to perform generalized moves, which look like R-moves but depend onw. Fig.6 shows some of them.

agb

cgd ε

b a

c d g

−1 g ε

a b g

ε

−

cg d−1 agb

ε

−

d c g

−1 −1

k

kf ch

k d eh b g

h a a

c

d e

f

h k

b g

Figure 6: Some generalized moves – for the R3picture, it is assumed thatghk= 1. Warning: the rules for the arrow orientations here depend on the value ofw(g).

Theorem 3.3. Let (Σ, x) be an arbitrary surface with a base point, and denote by (π, w) the weighted fundamental group of (Σ, x) (see Definition 2.2). There is a 1−1 correspondence Φ between Gauss diagrams on π up to R-moves and w-moves (i.e. virtual knot types on (π, w)), and virtual knot diagrams on Σ up to diagram isotopy, Reidemeister moves and detour moves (i.e. virtual knot types on Σ).

Proof. Fix a subsetX of Σ homeomorphic to a closed 2-dimensional disc and containing the base pointx– so thatπ=π1(Σ, X). Also,X being contractible allows one to fix a trivialization of the thickening of Σ overX: this gives meaning to the locallyover and under branches when a knot

diagram has a real crossing inX.

Construction of the bijection. Pick a knot diagram D ∈ Σ and assume that every real crossing ofD lies over X. ThenD defines a Gauss diagram on π, denoted by ϕ(D): the signs of the arrows are given by the writhes, their orientation is defined by the trivialization ofM →Σ over X, and each edge is decorated by the class inπof the corresponding arc inD. This defines ϕ(D) without ambiguity ifD has at least one real crossing. If it does not, then defineϕ(D) as a Gauss diagram without arrows, decorated with the conjugacy class corresponding to the free homotopy class ofD. Finally, put

Φ(D) := [ϕ(D)] mod R-moves andw-moves.

Invariance ofΦunder diagram isotopy and detour moves. It is clear from the definitions that ϕ(D) is strictly unchanged under detour moves on D. Now assume that D1 and D2 are equivalent underusual diagram isotopy – that is, diagram isotopy that may take real crossings out ofX for some time. By Corollary 2.7, it is enough to understand what happens for a diagram isotopy along which only one crossing goes out ofX. In that case,ϕ(D) is changed by aw-move performed on the arrow corresponding to that crossing, where the conjugating element g is the loop followed by the crossing along the isotopy. Indeed, since the first Stiefel-Whitney class of the thickening of Σ coincides with that of its tangent bundle, it follows that:

1. The orientation of the fibre (and thus the notions of “over” and “under”) is reversed alongg if and only ifw(g) =−1, which actually corresponds to the rule for arrow orientations in a w-move.

2. The orientation of the fibre over the crossing is reversed alongg if and only if a given local orientation of Σ is reversed alongg, so that the writhe of the crossing never changes.

Invariance ofΦunder Reidemeister moves. Up to conjugacy by a diagram isotopy, it can always be assumed that a Reidemeister move happens insideX. In that case, at the level ofϕ(D), it clearly corresponds to an R-move as described in Definition 3.1.

So far, Φ is a well-defined map from the set of virtual knot types on Σ to the set of virtual knot types on (π, w).

Construction of an inverse map Ψ. IfGis a Gauss diagram without arrows, then define ψ(G) as the totally virtual knot with free homotopy class equal to the marking of G– it is well- defined up to detour moves. If G has arrows, then for each of them draw a crossing inside X with the required writhe, and then join these by totally virtual arcs with the required homotopy classes. The resulting diagramψ(G) is well-defined up to diagram isotopy and detour moves by this construction. In both cases, put

Ψ(D) := virtual knot type ofψ(D).

Let us prove thatϕandψ are inverse maps, so that Ψ will be the inverse of Φ as soon as it is invariant under R-moves andw-moves.

It is clear from the definitions thatϕ◦ψcoincides with the identity. It is also clear that ψ◦ϕ is the identity, up to detour moves, fortotally virtual knot diagrams.

Now fix a knot diagram D with at least one real crossing (and all real crossings inside X).

Recall thatψ◦ϕ(D) is defined up to diagram isotopy and detour moves, so fix a diagramD^′ in that class. There is a natural correspondence between the set of real crossings of D and those ofD^′, due to the fact that both identify by construction with the set of arrows of ϕ(D). Pick a diagram isotopyhthat takes each real crossing of D to meet its match inD^′, without leaving X.

Then clearlyϕ(h(D)) =ϕ(D), and because ϕ◦ψis the identity, one gets

ϕ(h(D)) =ϕ(D^′). (1)

The choice ofhensures thath(D) and D^′ differ only by totally virtual arcs, and (1) implies that each of these, inh(D), has the same class inπ1(Σ, X) as its match inD^′, which means by definition

that h(D) and D^′ are equivalent up to detour moves. Thus ψ◦ϕis the identity up to diagram isotopy and detour moves.

Invariance of Ψ under R-moves. Let us treat only the case of R2-moves, which contains all the ideas. Let G1 and G2 differ by an R2-move, and assume that G1 is the one with more arrows. By appropriate diagram isotopy and detour movesinside X, performed on ψ(G1), it is possible to make the two concerned crossings “face” each other, as in Fig.7 (left). The pathsα1

and α2 from this picture are totally virtual and trivial in π1(Σ, X), thus ψ(G1) is equivalent to the second diagram of Fig.7 up to detour moves. The fact that at this point, an R-II move is actually possible is a consequence of (in fact equivalent to) the combinatorial conditions defining the R-moves. Denote byD the third diagram of the picture. The “merge multiply” principle that rules R2-moves implies thatϕ(D) =G2, so that

ψ(G1)∼D∼ψ◦ϕ(D) =ψ(G2), (2) where∼is the equivalence under diagram isotopy, detour moves and Reidemeister moves. It follows thatψ(G1) andψ(G2) have the same knot type.

α₂

α₁

X X X

Figure 7: R2-moves actually correspond to Reidemeister moves

X X

g

X

Figure 8: Performing a w-move – the railway trick

Invariance of Ψ under w-moves. LetG1 and G2 differ by aw-move on g∈π. Call c the corresponding crossing on the diagramψ(G1). Then, pick two little arcs right beforec, one on each branch, and make them followgby a detour move. At the end, one shall see a totally virtual 4-lane

railway as pictured on Fig.8 (middle): the strands are made parallel,i.e. any (virtual) crossing met by either of them is part of a larger picture as indicated by the zoom. This ensures that, using the mixed version of Reidemeister III moves, one can slide the real crossing all along the red part of the railway, ending with the diagram on the right of the picture – let us call itD. The conclusion is identical to that for R-moves: againϕ(D) =G2 and (2) holds, whenceψ(G1) andψ(G2) have the same knot type.

3.1.1 About the orbits ofw-moves

It could feel natural to try to get rid of w-moves by understanding their orbits in a synthetic combinatorial way. This is what is done in Section 3.2 in the particular case of an abelian group πendowed with the trivial homomorphismπ→F₂.

In general, for a Gauss diagram onπ, G, denote by h1(G) the set of free homotopy classes of loops in the underlying topological space ofG (it is the set of conjugacy classes in a free group on deg(G) + 1 generators). Also, denote by h1(π) the set of conjugacy classes in π. Then the π-markings ofGdefine a map

FG:h1(G)→h1(π).

Observe that the mapG7→FG is invariant underw-moves. This raises a number of questions that amout to technical group theoretic problems, and which will not be answered here (G^w denotes the orbit ofGunderw-moves):

1. Is the mapG^w7→FG injective?

2. If the answer to 1. is yes, then is G^w determined by a finite number of values of FG, for instance its values on the free homotopy classes ofsimple loops?

3. Is it possible to detect in a simple manner what maps h1(G)→ h1(π) lie in the image of G^w7→FG?

Remark 3.4. Gauss diagrams with decorations inh1(Σ) can be met for example in [11], where they are used to construct knot invariants in a thickened oriented surface Σ – see also Section 4.3. If the answer to Question 1. above is no, then such invariants, which factor throughFG, stand no chance to be complete.

Remark 3.5. Even for diagrams with only one arrow, it still does not seem easy to answer the

“simple loop” version of Question 2.Given x, y, h, kin a finite type free group, is it true that hxh⁻¹kyk⁻¹=xy =⇒ ∃l,

hxh⁻¹ = lxl⁻¹ kyk⁻¹ = lyl⁻¹ ?

Let us end with an example that shows that the values ofFGon the (finite) set of simple loops running along at most one arrow is not enough (cf. Question 2.). Fig.9 shows a Gauss diagram with such decorations –{a, b}is a set of generators for the free groupπ1(Σ)≃F(a, b), where Σ is a 2-punctured disc. These particular values ofFG do not determine the free homotopy class of the red loopγ, as it is shown in Fig.10.

In fact, these two virtual knots are even distinguished by Vassiliev-Grishanov’s planar chain invariants, which means they represent different virtual knot types.

3.2 Abelian Gauss diagrams

In this subsection, π is assumed to be abelian, and w0 denotes the trivial homomorphismπ → F₂. We describe a version of Gauss diagrams that carries as much information as the previously introduced virtual knot types on (π, w0), with two improvements:

➺ The diagrams are made of less data than in the general version.

➺ This version is free from conjugacy moves.

[a] [b]

+ γ [ab]

[b]

[a]

+

Figure 9: A Gauss diagram withh1-decorations that does not define a unique virtual knot

+ +

1

a b

1

+

+

−1 −1

a bab a b

1

Figure 10: One red loop is trivial, while the other is a commutator

It is inspired from the decorated diagrams introduced by T. Fiedler to study combinatorial invari- ants for knots in thickened surfaces (see [7, 8] and also [17]).

We use the same notationGfor a Gauss diagram and its underlying topological space, which has a 1-dimensional complex structure with edges and arrows as oriented 1-cells. H1(G) denotes its first integral homology group.

Definition-Lemma 3.6 (fundamental loops). Let G be a classical Gauss diagram of degree n.

There are exactlyn+ 1simple loops inGrespecting the local orientations of edges and arrows, and going along at most one arrow. They are called the fundamental loops of G and their homology classes form a basis ofH1(G).

Definition 3.7(abelian Gauss diagram). Letπbe an abelian group. Anabelian Gauss diagram onπis a classical Gauss diagramGdecorated with a group homomorphismµ:H1(G)→π. It is usually represented by its values on the basis of fundamental loops, that is, one decoration inπ for each arrow, and one for the base circle – that last one is called theglobal marking ofG.

A Gauss diagram onπdetermines an abelian Gauss diagram as follows:

➺ The underlying classical Gauss diagram is the same.

➺ Each fundamental loop is decorated by the sum of the markings of the edges that it meets (see Fig 11).

This defines anabelianization mapab.

Proposition 3.8. The map ab induces a natural 1−1 correspondence between abelian Gauss diagrams on π and equivalence classes of Gauss diagrams on π up to w0-moves. Moreover, if

a+b+c +d+e+f

a f

e

c d b

c

e+f b+c+d+e

ab

Figure 11: Abelianizing a Gauss diagram on an abelian group

π=π1(Σ)is the fundamental group of a surface, then these sets are in 1−1 correspondence with the set of virtual knot diagrams onΣup to diagram isotopy and detour moves.

Proof. The proof of the last statement is contained in that of Theorem 3.3 – through the facts that φandψare inverse maps up to detour moves and diagram isotopy, and thatw-moves at the level of knot diagrams can be performed using only detour moves and diagram isotopies, by the railway trick (Fig.8).

As for the first statement, one easily sees that ab is invariant under w0-moves. We have to show that conversely, if ab(G1) = ab(G2), thenG1andG2 are equivalent underw0-moves.

This is clear ifG1has no arrows, since then ab(G1) =G1. Now proceed by induction. SinceG1

andG2 have the same abelianization, they have in particular the same underlying classical Gauss diagram, and there is a natural correspondence between their arrows.

Case 1: No two arrows in G1 cross each other. Then at least one arrow surrounds a single isolated edge on one side (as in an R1-move). Choose such an arrow α and remove it, as well as its match in G2. By induction, there is a sequence of w0-moves on the resulting diagramG^′₁ that turns it intoG^′₂. Since the arrows ofG^′₁ have a natural match inG1, thosew0-moves make sense there, and take every marking ofG1 to be equal to its match inG2, except for those in the neighborhood ofα. So we may assume thatG1 and G2 only differ nearαas in Fig.12. Since all the unseen markings coincide inG1 and G2, and since ab(G1) and ab(G2) have the same global marking, it follows that

a+b+c=a^′+b^′+c^′.

Thus aw0-move onαwith conjugating elementg=a^′−aturnsG1intoG2. a

b

c

’

’

’ G2 a

b

c G1

α α

Figure 12: Notations for case 1

Case 2: There is at least one arrowαinG1that intersects another arrow. By the same process as in case 1, one may assume that G1 and G2 only differ near α– see Fig.13, where a, b, c and dactually correspond to pairwise distinct edges since αintersects an arrow. Again, since all the unseen markings coincide inG1andG2, one obtains

a+d=a^′+d^′, and

b+c=b^′+c^′,

by considering the global marking, and the marking ofα, in ab(G1) and ab(G2). Moreover, there is at least one arrow intersectingα: considering the marking of that arrow gives

a+b=a^′+b^′. The last three equations may be written as

a^′−a=b−b^′ =c^′−c=d−d^′,

so that, again, aw0-move on αwith conjugating elementg=a^′−aturnsG1 intoG2.

G2 G₁

α a b

c d

α b a

c d

’

’ ’

’

Figure 13: Notations for case 2

Remark 3.9. A different proof of this proposition was given in a draft paper, in the special case π=Z([16], Proposition 2.2). As an exercise, one can show that this proof extends to the case of an arbitrary abelian group.

To make the picture complete, it only remains to understand R-moves in this context.

Definition 3.10 (obstruction loops). Within any local Reidemeister picture like those shown on Fig.2 featuring at least one arrow, there is exactly one (unoriented) simple loop. We call it the obstruction loop. Fig.14 shows typical examples.

Definition 3.11(R-moves). A move from Fig.2 is likely to define an R-move only if the obstruction loop lies in the kernel of the decorating mapH1(G)→π(which makes sense even though the loop is unoriented). Under that assumption, theR-moves for abelian Gauss diagrams are defined by the usual conditions:

➺ i= 1.No additional condition.

➺ i= 2.The arrows head to the same edge, and have opposite signs.

➺ i= 3. The value of w(e)ε(e) is the same for all three visible edgese, and the values of ↑(e) are pairwise different (see Definition 1.2).

Theorem 3.12. The map ab induces a natural1−1 correspondence between equivalence classes of abelian Gauss diagrams onπ up to R-moves and virtual knot types on (π, w0).

Proof. ab clearly maps an R-move in the non commutative sense to an R-move in the abelian sense.

Conversely, if ab(G1) and ab(G2) differ from an (abelian) R-move, then the vanishing homological obstruction implies thatG1 andG2 are in a position to perform a “generalized R-move” like the examples pictured on Fig.6.

Theorems 3.3 and 3.12 together imply the following

Corollary 3.13. IfΣis an orientable surface with abelian fundamental group, then there is a1−1 correspondence between abelian Gauss diagrams on π1(Σ) up to R-moves, and virtual knot types onΣ.

R 2 R 1

R 3

Figure 14: Homological obstruction to R-moves

3.3 Homological formulas

It may seem not easy to compute an arbitrary value of the linear map decorating an abelian Gauss diagram, given only its values on the fundamental loops. To end this section, we give two formulas to fill this gap, by understanding the coordinates of an arbitrary loop in the basis of fundamental loops.

3.3.1 The energy formula

Fix an abelian Gauss diagramG. Observe that as a cellular complex,Ghas no 2-cells, thus every 1-homology class has a unique set of “coordinates” along the family of edges and arrows. For each 1-cell c (which may be an arrow or an edge), we denote by h·, ci : H1(G) → Z the coordinate function alongc. It is a group homomorphism.

Let us denote by [A] ∈H1(G) the class of the fundamental loop associated with an arrowA (Fig.15 left).

Definition-Lemma 3.14 (Energy of a loop). Fix an edge e in G, and a class γ ∈H1(G). The value of

Ee(γ) =hγ, ei − X

h[A],ei=1

hγ, Ai (3)

is independent ofe. This defines a group homomorphism E:H1(G)→Z.

Proof. Let us compare the values ofE·(γ) for an edgeeand the edgee^′ right after it. eande^′are separated by a vertexP, which is the endpoint of an arrowA. There are two possible situations (Fig.15):

1. P is the tail ofA. Thenh[A], ei= 1 and h[A], e^′i= 0, so that Ee(γ)−Ee^′(γ) =hγ, ei − hγ, Ai − hγ, e^′i. 2. P is the head ofA. Thenh[A], ei= 0 andh[A], e^′i= 1, so that

Ee(γ)−Ee^′(γ) =hγ, ei+hγ, Ai − hγ, e^′i. In both cases,Ee(γ)−Ee^′(γ) is equal to h∂γ, Pi, which is 0 sinceγis a cycle.

Theorem 3.15. For anyγ∈H1(G), one has the decomposition γ=X

A

hγ, Ai[A] +E(γ) [K]. (4)

e e’

A [A] A

e e’

[A] A [A]

Figure 15: The fundamental loop of an arrow and the two cases in the proof of Lemma 3.14 Proof. This formula is an identity between two group homomorphisms, so it suffices to check it on the basis of fundamental loops, which is immediate.

Remark 3.16. The existence of a map E such that Theorem 3.15 holds was clear, since for each arrowAconsidered as a 1-cell, [A] is the only fundamental loop that involvesA. With that in mind, one may read into (3) as follows: E(γ) counts the (algebraic) number of times thatγgoes through an edge, minus the number of those times that are already taken care of by the fundamental loops of the arrows. This number has to be the same for all edges, so that one recovers a multiple of [K].

3.3.2 The torsion formula

Looking at (4) and Fig.15, one may feel that it would be more natural to have [K]−[A] involved in the formula, instead of [A], for all arrowsAsuch thathγ, Aiis negative – that is, whenγ runs alongA with the wrong orientation more often than not. The formula then becomes

γ= X

hγ,Ai>0

hγ, Ai[A] + X

hγ,Ai<0

hγ, Ai([A]−[K]) − T(γ) [K], (5) where

− T(γ) =E(γ) + X

hγ,Ai<0

hγ, Ai. (6)

Definition 3.17. T(γ) is called thetorsion ofγ.

How is (5) different from (4)?

⊖On the negative side, unlike the energy,T is not a group homomorphism. But it actually behaves almost like one:

Lemma 3.18. Letγ1 andγ2 be two homology classes such that

∀A, hγ1, Ai hγ2, Ai ≥0.

Then

T(γ1+γ2) =T(γ1) +T(γ2).

Proof. It follows from the definition and the fact thatE(γ) is a homomorphism.

⊕On the positive side:

Lemma 3.19. The torsion of a loop in a Gauss diagram Gdoes not depend on the orientations of the arrows ofG.

Proof. By expanding the defining formula, T(γ) =− hγ, ei + X

hγ, Ai<0 h[A], ei= 0

hγ, Ai − X

hγ, Ai>0 h[A], ei= 1

hγ, Ai,

one sees that reversing an arrow makes its contribution (if non zero) switch from one sum to the other, whilehγ, Aialso changes signs.

This lemma allows one to expect that T(γ) should admit a very simple combinatorial inter- pretation. It actually does, but only for a certain family of loops – the ERS loops defined below.

Fortunately enough, this family happens to positively generateH1(G), which allows one to compute the torsion of any loop by using Lemma 3.18.

Definition 3.20. The notation γ is used for loops as well as 1-homology classes. A homology classγ∈H1(G) is said to be

➺ ER(for “edge-respecting”), if for every edgee,hγ, ei ≥0.

➺ simpleif it is the class of a simple (injective) loop, that is,|hγ, ci| ≤1 for every 1-cellc(edge or arrow).

➺ ERS if it is ER and simple.

➺ proper if it runs along at least one arrow.

(a) (b)

γ

Figure 16: The local and global look of a proper ERS loop Consider a permutationσ∈S(J1, nK), and set

ր(σ) :=♯{i∈J1, nK|σ(i)> i}. It is easy to check that ifσ0is the circular permutation (1 2. . . n), then

∀σ∈S,ր(σ) =ր(σ0σσ⁻¹₀ ).

Definition 3.21. The invariance property from above means that T is well-defined for permuta- tions of a set ofnpoints lying in an abstract oriented circle. We still denote this function byT, and call it thetorsion of a permutation.

Let γ be a proper simple loop, then the set of edges e such that hγ, ei 6= 0 can be naturally assimiliated to a finite subset of an oriented circle, andγ induces a permutation of this set. Let us denote it byσγ.

Theorem 3.22. For all proper ERS loopsγ,

T(γ) =ր(σγ).

This theorem can be useful in practice, since the torsion of a permutation can be computed at a glance on the braid-like presentation. Observe that

1. Every non proper loop is homologous to a multiple of [K], easy to determine.

2. For every proper loopγ, there is an integernsuch that ˜γ=γ+n[K] is proper, ER, and has zero coordinate along at least one edge. Namely,n=−minehγ, ei.

3. Every class ˜γas above may be decomposed as a sum ˜γ=P

iγi such that

➺ all theγi’s are proper and ERS

➺ ∀i, j, A,hγi, Ai hγj, Ai ≥0 4. By Lemma 3.18,T(˜γ) =P

iT(γi), and theT(γi)’s are given by Theorem 3.22.

This shows that it is possible to compute any homology class by using the torsion formula.

Whether it is more interesting than the energy formula depends on the context.

Proof of Theorem 3.22. One may assume that for every arrowA, hγ, Ai= 1.Indeed, deleting an arrow avoided byγ, or reversing the orientation of an arrow thatγruns in the wrong direction, have no effect on either side of the formula (notably because of Lemma 3.19). Under this assumption, half of the edges ofGare run by γ: call them the red edges ofG, while the other half are called theblue edges. Red and blue edges alternate along the orientation of the circle.

Ifeis any (red or blue) edge, we define:

λ(e) :=X

A

h[A], ei.

Lemma 3.23. Under the assumption that hγ, Ai= 1 for allA, the value of λ(e)only depends on the color of the edgee. Moreover,

λ(blue) =λ(red)−1 =ր(σγ).

Let us temporarily admit this result. By the definition ofλ, P

A[A] = P

arrows +λ(red)P

(red edges) +λ(blue)P

(blue edges)

Lemma3.23

= P

arrows +P

(red edges) +λ(blue)P

(red and blue edges)

= γ+λ(blue)[K]

Lemma3.23

= γ+ ր(σγ)[K].

Since it was assumed thathγ, Ai= 1 for every arrow, the definition ofT (5) reads γ=X

A

[A]− T(γ)[K], which terminates the proof of the theorem, up to Lemma 3.23.

Proof of Lemma 3.23. In the case of σ0 = (1 2. . . n) depicted on Fig.17, it is easy to see that λ(red) =nand λ(blue) =n−1, while ր(σ0) = n−1. The lemma being true for one diagram, let us show that it survives elementary changes that cover all the diagrams.

σ

σ

. . .

. . .

Figure 17: Braid-like representations of permutations are to be read from bottom to top Notice that for every proper ERS loopγ,σγ is a cycle, and conversely a permutation that is a cycle uniquely defines an undecorated Gauss diagram and a proper ERS loopγ such that for every arrowA, hγ, Ai= 1. Thus, covering all possible permutations implies covering all possible diagrams and proper ERS loops. So all we have to check is that the formula survives an operation onσγ, of the form:

(. . . i j . . .)−→(. . . j i . . .)

(... j i ...) (... i j ...)

i j

i j

i j

σ ∼ (i)=j

σ (i)=j σ (i)=j

σ σ

(j)=i σ

(j)=i

σ σ∼

∼

(j)=i

∼

j i j i

j i i j

i j i j

type A

type B

type C

Figure 18: Twist moves on Gauss diagrams

The corresponding move at the level of Gauss diagrams may be of six different types, grouped in three pairs of reverse operations (Fig.18).

On each diagram in Fig.18, the three moving arrows split the base circle into six regions. One computes the variation ofλ separately for each of these regions, and sees that it is the same for each of them. The results are gathered in the following table, proving the lemma.

type of move variation ofλ variation ofT(γ)

A unchanged unchanged

B (from left to right) decreases by 1 decreases by 1 C(from left to right) decreases by 1 decreases by 1

4 Finite-type invariants

One of the main points of using Gauss diagrams is their ability to describe finite-type invariants by simple formulas [20, 7, 2, 3]. In the case of classical long knots in 3-space, such formulas actually cover all Vassiliev invariants as was shown by M.Goussarov [9]. In the virtual case, the two notions actually differ (see [13] and also [5, 4]. Finite type invariants for virtual knots that do admit Gauss diagram formulas shall be called GPV invariants [10].

In [17], a simple set of criteria was given to detect a particular family of those formulas, called virtual arrow diagram formulas. Most of the examples that are known belong to this family.

That includes Chmutov-Khoury-Rossi’s formulas for the coefficients of the Conway polynomial [2]

(and their generalization by M. Brandenbursky [1]), as well as the formulas from [7, 8, 11] where different kinds of decorated diagrams are used. Note however that the formulas for the invariants extracted from the HOMFLYPT polynomial [3] are arrow diagram formulas only if the variablea is specialized to 1 (which yields back the result of [2]).

In this section, we extend the results from [17] to an arbitrary surface. Then we show how to apply them to any other kind of decorated diagrams found in the literature, by definingsymmetry- preserving mapswhich enable one to jump from one theory to another.

4.1 General algebraic settings

We denote by G_n (resp. G_≤n) the Q-vector space freely generated by Gauss diagrams on π of degreen(resp. ≤n), and setG= lim−→G_≤n. Unlessπis a finite group, these spaces are not finitely generated, and we define their hat versions Gb_n (resp. Gb_≤n) as the Q-spaces of formal series of Gauss diagrams of degreen(resp. ≤n). Finally, set Gb = lim−→Gb_≤n.An arbitrary element ofGb is usually denoted byG and called aGauss series, of degreenif it is represented inGb_≤n but not in b

G_≤n−1. The notationGis saved for single Gauss diagrams.

A Gauss diagramG of degree n has a group of symmetries Aut(G), which is a subgroup of Z/2n, made of the rotations of the circle that leave unchanged a given representative of G(see Subsection 4.1.2). Gis endowed with the orthonormal scalar product with respect to its canonical basis, denoted by (,), and its normalized versionh,i, defined by

hG, G^′i:=|Aut(G)|(G, G^′). (7) There is a linear isomorphismI:G_≤n →G_≤n, the keystone to the theory, which maps a Gauss diagram of degreento the formal sum of its 2ⁿ subdiagrams:

I(G) = X

σ∈{±1}ⁿ

G(σ), (8)

whereG(σ)is Gdeprived from the arrows that σmaps to−1 (see Definition3.1 for subdiagrams).

The inverse map ofI is given by

I⁻¹(G) = X

σ∈{±1}ⁿ

sign(σ)G(σ). (9)

Definition 4.1. A finite-type invariant for virtual knots in the sense of Goussarov-Polyak-Viro is a virtual knot invariant given by aGauss diagram formula

νG :G7→ hG, I(G)i, (10)

whereG ∈Gb. Such a formula “counts” the subdiagrams ofG, with weights given by the coefficients ofG. Notice that only one of the two arguments ofh,ineeds to be a finite sum for the expression to make sense. We do not make a distinction between a virtual knot invariant and the linear form induced onG.

4.1.1 The Polyak algebra

A Gauss seriesG ∈Gb defines a virtual knot invariant if and only if the functionhG, I(.)iis zero on the subspace spanned by R-moves andw-moves relators. Hence one has to understand the image of that subspace underI with a simple family of generators. This is the idea of the construction of the Polyak algebra ([19, 10]).

In the present case,P is defined as the quotient ofGby

➺ the relations shown in Fig.19, which we call P1, P2, P3 (or 8T relation),

➺ the W relation, which is simply the linear match ofw-moves (i.e. just replace the “!” with a “=” in all the relations from Fig.5).

Be careful that unlike R1-moves, where an isolated arrow surrounding an edge marked with 1 simply disappears, in a P1-move the presence of such an arrow completely kills the diagram. Fig.19 does not feature theπ-markings for P3 to lighten the picture, but they have to follow the usual

“merge multiply” rule (see Definition 3.1).

The following proposition extends Theorem 2.Dfrom [10].

Proposition 4.2. The map I induces an isomorphism G/R,W→G/P,W =:P. More precisely, I induces an isomorphism between Span(Ri)and Span(Pi), for i= 1,2,3, and between Span(W) and itself. It follows that the mapG→I(G)∈ P defines a complete invariant for virtual knots.

"8T"

or

P 3

=

1

1 1 1

1

1 1 1

1

1 1 1

+ +

−

+ + +

+ + +

+− +

− +

+

+

+

− +

− + +−

+ ε

2 −

P a _−ε

d c

b

c

b a

d c

b

+ +

1 a

d

0

ε

1

ε

=

ε

− −ε

a

d c

b a

d c

b a

d c

b

ε

1

0

+ +

1

ε

=

1 0 P 1

=

Figure 19: The three kinds of Polyak relations – only one P3 relation is shown, there is a second one obtained by reversing all the arrow orientations.

4.1.2 The symmetry-preserving injections

Depending on the context, one may have to consider simultaneously different types of Gauss diagrams, with more or less decorations. This subsection presents a natural way to do it, convenient from the viewpoint of Gauss diagram invariants. The construction requires one to choose a kind of combinatorial objects that is the “father” of all other kinds, in the sense of quotienting. We present the construction by taking as the father type that of Gauss diagrams on a group.

In first place, we do not regard Gauss diagrams up to homeomorphisms of the circle: the base circle is assumed to be the unit circle inC, the endpoints of the arrows are assumed to be located at the 2n-th roots of unity, and the arrows are straight line segments. Such a diagram is called rigid.

By a “type of rigid Gauss diagrams” we mean an equivalence relation on the set of rigid Gauss diagrams onπ, which is required to satisfy two properties:

1. (Degree property) All diagrams in a given equivalence class shall have the same degree.

2. (Stability property) The action ofZ/2non the set of Gauss diagrams of degreenshall induce an action on the set of degreenequivalence classes.

Since every construction in this subsection is therefore destined to be homogeneous, the degree of all Gauss diagrams is once and for all set equal to n. Arigid Gauss diagram of type ∼ is an equivalence class under the relation∼. AGauss diagram (of type∼)is the orbit of a rigid diagram of type∼under the action ofZ/2n. The correspondingQ-spaces are respectively denoted byG^rigid

∼

andG_∼.

SinceZ/2nis abelian, two elements from the same orbit have the same stabilizer, hence a Gauss diagramGhas a well-definedgroup of symmetriesAut(G), which is the stabilizer of any of its rigid representatives under the action ofZ/2n. Consequently, the spaceG_∼ is endowed with a pairing h,idefined by (7).

Now consider two types of rigid Gauss diagrams, say 1 and 2, such that relation 1 is finer than relation 2 (“1≺2”).